{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:17:33Z","timestamp":1760239053376,"version":"build-2065373602"},"reference-count":12,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2020,9,25]],"date-time":"2020-09-25T00:00:00Z","timestamp":1600992000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>For the associated Legendre and Ferrers functions of the first and second kind, we obtain new multi-derivative and multi-integral representation formulas. The multi-integral representation formulas that we derive for these functions generalize some classical multi-integration formulas. As a result of the determination of these formulae, we compute some interesting special values and integral representations for certain particular combinations of the degree and order, including the case where there is symmetry and antisymmetry for the degree and order parameters. As a consequence of our analysis, we obtain some new results for the associated Legendre function of the second kind, including parameter values for which this function is identically zero.<\/jats:p>","DOI":"10.3390\/sym12101598","type":"journal-article","created":{"date-parts":[[2020,9,28]],"date-time":"2020-09-28T10:39:58Z","timestamp":1601289598000},"page":"1598","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Multi-Integral Representations for Associated Legendre and Ferrers Functions"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9398-455X","authenticated-orcid":false,"given":"Howard S.","family":"Cohl","sequence":"first","affiliation":[{"name":"Applied and Computational Mathematics Division, National Institute of Standards and Technology, Mission Viejo, CA 92694, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9545-7411","authenticated-orcid":false,"given":"Roberto S.","family":"Costas-Santos","sequence":"additional","affiliation":[{"name":"Departamento de F\u00edsica y Matem\u00e1ticas, Universidad de Alcal\u00e1, c.p. 28871 Alcal\u00e1 de Henares, Spain"}]}],"member":"1968","published-online":{"date-parts":[[2020,9,25]]},"reference":[{"key":"ref_1","first-page":"14","article-title":"Opposite Antipodal Fundamental Solution of Laplace\u2019s Equation in Hyperspherical Geometry","volume":"7","author":"Cohl","year":"2011","journal-title":"Symmetry Integr. Geom. Methods Appl."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"145206","DOI":"10.1088\/1751-8113\/45\/14\/145206","article-title":"Fourier and Gegenbauer expansions for a fundamental solution of the Laplacian in the hyperboloid model of hyperbolic geometry","volume":"45","author":"Cohl","year":"2012","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_3","unstructured":"Olver, F.W.J., Olde Daalhuis, A.B., Lozier, D.W., Schneider, B.I., Boisvert, R.F., Clark, C.W., Miller, B.R., Saunders, B.V., Cohl, H.S., and McClain, M.A. (2020, September 15). NIST Digital Library of Mathematical Functions, Available online: http:\/\/dlmf.nist.gov\/."},{"key":"ref_4","unstructured":"Abramowitz, M., and Stegun, I.A. (1972). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables."},{"key":"ref_5","unstructured":"Erd\u00e9lyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F.G. (1981). Higher Transcendental Functions. Volume I, Robert E. Krieger Publishing Co. Inc."},{"key":"ref_6","first-page":"45","article-title":"Fundamental Solutions and Gegenbauer Expansions of Helmholtz Operators in Riemannian Spaces of Constant Curvature","volume":"14","author":"Cohl","year":"2018","journal-title":"Symmetry Integr. Geom. Methods Appl. (SIGMA)"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Andrews, G.E., Askey, R., and Roy, R. (1999). Special Functions, Encyclopedia of Mathematics and Its Applications; Cambridge University Press.","DOI":"10.1017\/CBO9781107325937"},{"key":"ref_8","unstructured":"Olver, F.W.J. (1997). Asymptotics and Special Functions, AKP Classics, A K Peters Ltd.. Reprint of the 1974 original [Academic Press, New York]."},{"key":"ref_9","first-page":"16","article-title":"On parameter differentiation for integral representations of associated Legendre functions","volume":"7","author":"Cohl","year":"2011","journal-title":"Symmetry Integr. Geom. Methods Appl."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Magnus, W., Oberhettinger, F., and Soni, R.P. (1966). Formulas and Theorems for the Special Functions of Mathematical Physics, Springer. [3rd ed.]. Die Grundlehren der mathematischenWissenschaften, Band 52.","DOI":"10.1007\/978-3-662-11761-3"},{"key":"ref_11","unstructured":"Gradshteyn, I.S., and Ryzhik, I.M. (2007). Table of Integrals, Series, and Products, Elsevier\/Academic Press. [7th ed.]."},{"key":"ref_12","unstructured":"Srivastava, H.M., and Karlsson, P.W. (1985). Multiple Gaussian Hypergeometric Series, Ellis Horwood Ltd."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/10\/1598\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:13:47Z","timestamp":1760177627000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/10\/1598"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,9,25]]},"references-count":12,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2020,10]]}},"alternative-id":["sym12101598"],"URL":"https:\/\/doi.org\/10.3390\/sym12101598","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2020,9,25]]}}}